Compressed sensing using regional sparsity

ABSTRACT

A method for reconstructing an image includes acquiring raw image data during a scan of an area, estimating an image from the raw image data, separating the estimated image into a region of interest (ROI) and a background region, and applying compressed sensing to iteratively update only the ROI and maintain the background region to reconstruct an image.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application No.61/555,172 filed on Nov. 3, 2011, the disclosure of which isincorporated by reference herein.

BACKGROUND OF THE INVENTION

1. Technical Field

The present disclosure relates generally to acquisition of imagingsignal and reconstruction of images from the acquired signals, and moreparticularly to acquisition and reconstruction of images usingcompressed sensing techniques.

2. Discussion of Related Art

Magnetic resonance imaging (MRI) is a non-invasive diagnostic imagingprocedure that uses nuclear magnetization and radio waves to produceinternal images of a patient. An MRI scanner contains magnetic coilsthat create a strong static magnetic field in which the patient ispositioned. Certain atoms in a patient's body that were previouslyrandomly-ordered become aligned along the magnetic field. The scannerthen sends a series of bursts or pulses of radio frequency (RF) energythrough the patient's body part under examination that excite the“ordered” atoms to specific oscillations around the magnetic field. Theatoms generate an RF signal during the pulsed oscillations and as theatoms return to their respective alignments. The scanner detects the RFsignals by appropriate reception or pick-up coils and uses gradientcoils to generate non-homogeneous magnetic fields to enable the signalsto be spatially coded in all three spatial directions. The scannerprocesses the coded signals or data to transform them into a visualrepresentation of the scanned patient's body part. In particular, thescanner samples and digitizes the signals, creates a so-called k-spacedata matrix filled with the digitized complex values of each signal, andgenerates for display and/or other usage a corresponding MR image fromthe k-space data matrix by means of a complex Fourier transform. The MRIscanner acquires three-dimensional image data of the patient's body partfor respective “slices” of an area of the body part. The scanner repeatsa pre-defined MR image pulse sequence, i.e., the above-described stepsfor collecting the signals/data, a number of times to collect sufficientdata from the excitations to reconstruct the specific image. Ideally,there are little or no variations in the nuclear magnetization duringthe excitations.

Image acquisition may be performed under time-sensitive conditions toensure that there is no movement of the subject during the imageacquisition process. Thus, image acquisition may be performed while thepatient refrains from moving. Often this is requires the patient holdinghis breath. When the MRI is used to track cardiac motion, acquisitiontime needs to be quick.

In light of this shortened acquisition time, it may be difficult toacquire data at the Nyquist rate to ensure sufficient sampling for idealimage reconstruction. Accordingly, performing accurate reconstructionwith less than an ideal amount of data may be difficult. This difficultyin reconstructing an image under these conditions may be similar totrying to solve for a system of linear equations in which there are moreunknown variables then there are equations. In such a case, there may bean infinite number of possible solutions.

Compressed sensing (CS) techniques have been developed to aid inreconstructing a signal using a sampling rate that is below the Nyquistsampling rate. These techniques exploit the observation that mostpractical signals of interest have sparse representations using aspecific transform. Thus, for a given signal, there may exist aparticular transform space in which a majority of the transformcoefficients are at or near zero. This transform space may be referredto as the sparsity space. As these small coefficients may be assumed tobe zero without significant loss of signal quality (the sparsenessassumption), signal reconstruction may be approximated by determiningonly the limited set of large transform coefficients for the sparsityspace.

In standard MRI acquisition, oversampling may be performed along afrequency encoding (or readout) direction to avoid aliasing (orwrapping) artifacts. In radial trajectory acquisition, the oversamplingmay be performed in each “readout” direction of radial spokes. Thisoversampling is usually not an issue for a modern scanner and does notincrease the total reconstruction time. However, for CS reconstruction,it is quite time-costly due to non-linear iterative optimizationinvolved to process the extra oversampled data. The time spent onreconstructing the peripheral background is considered time wasted sincein most cases they are zeros or close to zero (e.g., background noise),and will be thrown away at the end of reconstruction.

Thus, there is a need for methods and systems that can improve CSreconstruction.

SUMMARY OF THE INVENTION

According to an exemplary embodiment of the invention, a method forreconstructing an image includes estimating an image from acquired rawimage data, separating the estimated image into a region of interest(ROI) and a background region; and applying compressed sensing toiteratively update only the ROI and maintain the background region toreconstruct an image.

In an embodiment, the background entirely surrounds the ROI.

In an embodiment, the separating further separates the estimated imageinto a second ROI distinct from the first ROI, and the applyingcomprises applying the compressed sensing to iteratively update the ROIusing a first sparsity constraint and iteratively update the second ROIusing a second other sparsity constraint.

According to an exemplary embodiment of the invention, a method forreconstructing an image includes generating a cost function including afidelity term and a sparsity term and minimizing the generated costfunction to generate a reconstructed is image. The fidelity termincludes a difference between a transform of an estimated image and rawimage data. The sparsity term includes a mask multiplied by theestimated image that retains a region of interest (ROI) of the estimatedimage and masks out a background of the estimated image.

BRIEF DESCRIPTION OF THE DRAWINGS

Exemplary embodiments of the invention can be understood in more detailfrom the following descriptions taken in conjunction with theaccompanying drawings in which:

FIG. 1 illustrates a method of performing image reconstruction accordingto an exemplary embodiment of the invention;

FIG. 2 illustrates an example of a region of interest (ROI) and abackground of the ROI that may be used in reconstruction of an imageaccording to the method of FIG. 1,

FIGS. 3 a, 3 b, 3 c, 3 d illustrate a comparison between resultsobtained using a conventional CS image reconstruction method withoutseparating different regions of image and a CS image reconstructionmethod according to an exemplary embodiment of the invention;

FIG. 4 illustrates a method of reconstructing an image according to anexemplary embodiment of the invention;

FIG. 5 illustrates a CS approach according to an exemplary embodiment ofthe inventive;

FIG. 6 illustrates an example of a computer system capable ofimplementing methods and systems according to embodiments of the presentinvention.

DETAILED DESCRIPTION

Exemplary embodiments of the invention are discussed in further detailwith reference to FIGS. 1-6. This invention may, however, be embodied indifferent forms and should not be construed as limited to theembodiments set forth herein.

It is to be understood that the systems and methods described herein maybe implemented in various forms of hardware, software, firmware, specialpurpose processors, or a combination thereof. In particular, at least aportion of the present invention may be implemented as an applicationcomprising program instructions that are tangibly embodied on one ormore program storage devices (e.g., hard disk, magnetic floppy disk,RAM, ROM, CD ROM, etc.) and executable by any device or machinecomprising suitable architecture, such as a general purpose digitalcomputer having a processor, memory, and input/output interfaces. It isto be further understood that, because some of the constituent systemcomponents and process steps depicted in the accompanying Figures may beimplemented in software, the connections between system modules (or thelogic flow of method steps) may differ depending upon the manner inwhich the present invention is programmed. Given the teachings herein,one of ordinary skill in the related art will be able to contemplatethese and similar implementations of the present invention.

FIG. 1 illustrates a method of performing CS reconstruction according toan exemplary embodiment of the invention. Referring to FIG. 1, themethod includes: acquiring raw image data during a scan of an area(S101). The scanned area could be various objects such as a human body,an animal body, a shipping container, luggage, etc.

For example, one of various imaging scanning modalities can be used toscan the area to acquire the raw image data, such as MRI, computedtomography (CT), positron emission tomography (PET), etc.

In MRI, the raw image data may correspond to the 2D or 3D Fouriertransform of the MR image measured (i.e., k-space). Its complex valuesare sampled during an MR measurement, in a premeditated schemecontrolled by a pulse sequence, i.e., an accurately timed sequence ofradiofrequency and gradient pulses. When the k-space is full (e.g., atthe end of a scan), the data is ready to be processed to produce a finalimage. The raw image data may be acquired using Cartesian ornon-Cartesian acquisition schemes. When the non-Cartesian acquisitionscheme is used, the resulting raw image data may be gridded data.

Referring back to FIG. 1, the method includes estimating an image fromthe raw image data (S102). For example, if the area had been scannedusing MRI, an inverse Fourier transform can be performed on the k-spacedata to generate the estimated image. However, if the area had beenscanned using CT, a back-projection technique could have been used togenerate the estimated image. For other imaging modalities, differenttechniques can be used to generate the estimated image.

Had the raw image data been acquired using MRI and a non-Cartesiancoordinate system, the estimation of the image could be performed byperforming an inverse Fourier transform on the gridded k-space data.

Next, the method includes separating the estimated image into a regionof interest (ROI) and a background region (S103). In an exemplaryembodiment, the ROI is a center region and the background region is aperipheral region entirely surrounding the center region. However, theinvention is not limited to any particular location or shape for the ROIand the background region. Masks can be applied to the estimated imageto filter out the ROI and the background region. The use of these maskswill be described in more detail below. Next, the method includesapplying compressed sensing (CS) to iteratively update only the ROI andmaintain the background region to reconstruct an image (S104).

CS requires (1) signals or images to be sparse or compressible; (2) arandom sampling scheme; and (3) a non-linear optimization scheme toreconstruct the images. However, a truly random sampling of the k-space(e.g., spatial frequency space when MRI is used) of images may beimpractical. Both Cartesian and non-Cartesian acquisition schemes can beused to pseudo-randomly sample the k-space. However, a non-Cartesianacquisition scheme may achieve higher randomness than the Cartesianscheme. A radial trajectory acquisition is an example of thenon-Cartesian acquisition scheme.

In an MRI radial trajectory acquisition, MR raw age data (e.g., datapoints having frequency/phase coordinates) is acquired along radialspokes. As discussed above, the acquired MR raw image data may bereferred to as k-space data. However, since the k-space data capturedduring radial trajectory acquisition is along radials, their coordinatesare typically non-integer values. Integer valued coordinates are thenobtained using gridding since they are required for a Fast Fouriertransform (FFT). Gridding is a method of interpolating data to a uniformgrid, which can be used to generate gridded k-space data.

CS, also known as compressive sensing, compressive sampling, and sparsesampling, is a technique for finding sparse solutions to underdeterminedlinear systems. An underdetermined system of linear equations has moreunknowns than equations and generally has an infinite number ofsolutions. However, if there is a unique sparse solution to theunderdetermined system that presents sparsity in a certain space, thenthe CS framework allows the recovery of that solution. CS typicallystarts with taking a weighted linear combination of samples (compressivemeasurements) in a basis different from the basis in which the signal isknown to be sparse. Therefore, the task of converting the image backinto the intended domain involves solving an underdetermined matrixequation. By adding the sparsity constraints, even the number ofcompressive measurements taken is smaller than the number of pixels inthe full image.

Use of CS to reconstruct an image is an optimization problem. Accordingto an exemplary embodiment of the invention, CS can reconstruct an imageby minimizing the below cost function shown by the following equation 1:

ƒ({circumflex over (m)})=∥ℑ{circumflex over (m)}−d∥ ₂ +λ∥Ψ{circumflexover (m)}∥ ₁   (1)

where {circumflex over (m)} is an estimated image, Ψ is a sparsitytransform, d is gridded measured/acquired k-space data, ℑ is a Fouriertransform, and λ is a weighting factor. However, in an alternateembodiment, the gridded k-space data ‘d’ is replaced with the k-spacedata (i.e., the k-space data is not gridded). When an image modalityother than MRI is used, the ℑ stands for the corresponding transform ortechnique that is used with that modality (e.g., back-projection withCT). The sparsity transform Ψ applies a transform to the estimated image{circumflex over (m)} to yield a sparse result. In an embodiment, thesparsity transform Ψ is a spatial total variance and is applied to theestimated image {circumflex over (m)}.

The acquiring step S101 may perform an oversampling during itsacquisition of the raw image data. Oversampling is the process ofsampling a signal with a sampling frequency higher than twice thebandwidth or highest frequency of the signal being sampled. However,this oversampling can result in a doubled field of view (FOV) in theoversampling directions (e.g., in the radial sampling case, 4 timeslarger than the defined FOV) where the ROI occupies only part of theentire sampled region while the remaining “background” region occupiesthe remainder of the sampled region. The “background” region is thenchopped off and only the ROI is maintained as reconstruction results.For Cartesian acquisition, the oversampling can result in a doubled FOVin the frequency direction.

FIG. 2 illustrates an example of a ROI 201 and a background region 202due to oversampling that may be used in the reconstruction method ofFIG. 1. While FIG. 2 illustrates the ROI 201 as a “Center” region andthe background region 202 as a “Peripheral” region surround the “Centerregion”, embodiments of the invention are not limited thereto. Forexample, the background region 202 need not entirely surround the ROI201, and the ROI 201 can have various shapes and location within thefield of view.

While extra oversampled data can be handled by a fast Fourier transformreconstruction, it is extremely computation-expensive for CSreconstruction, which involves iterative calculations to solve the abovenon-linear optimization problem shown in equation 1.

The time spent on reconstructing the peripheral background is consideredtime wasted since in most cases they are zeros or close zeroes(background noise) and will be thrown away at the end of thereconstruction.

To effectively reduce the amount of data to be processed in order todecrease reconstruction time, an exemplary embodiment of the inventionapplies a regional sparsity approach into CS reconstruction. Asdiscussed above with respect to step

S103, the estimated image is separated into two parts, its ROI and a“background” region. For example, the ROI is distinct from thebackground region. Then as discussed above with respect to step S104,only the ROI is calculated and updated through the CS calculation whilethe “background region” (e.g., due to oversampling) is kept the same asthe initial estimation of the image. However, all the measured k-spacedata is still used so all the acquired “true” information is keptthrough the reconstruction.

The below describes how the regional sparsity approach is applied togenerate a modified version of the cost function shown in equation 1,according to an exemplary embodiment of the invention.

The estimated image {circumflex over (m)} in the cost function can bereplaced by the terms on the right side of equation 2 according to anexemplary embodiment as follows:

{circumflex over (m)}=W _(c) {circumflex over (m)}+W _(p) {circumflexover (m)} ₀   (2)

where W_(c) and W_(p) are ROI and “background” masks respectively, withW_(c)∩W_(p)=Θ and W_(c)∩W_(p)=I. In this way, instead of a currentestimate of an image being based on its ROI and background parts, it isbased on its ROI part and the background part of its initial estimate.In an embodiment, the ROI mask W_(c) is a matrix of 1s and 0s, where the0s correspond in location to the background region and the 1s correspondin location to the ROI. For example, when the ROI mask W_(c) ismultiplied by an estimated image, the ROI of the image is retained, andthe background part of the image is filtered out. In an embodiment, thebackground mask W_(p) is a matrix of 1s and 0s, where the 1s correspondin location to the background region and the 0s correspond in locationto the ROI. For example, when the background mask W_(p) is multiplied byan estimated image, the background part is retained and the ROI part isfiltered out.

Due to equation 2, during an initial iteration of minimizing the costfunction of equation 1, a calculation is performed that includes amultiplying the background mask W_(p) by the initial estimated image{circumflex over (m)}₀ to generate a second term, multiplying the centermask W_(c) by the initial estimated image {circumflex over (m)}₀ togenerate a first term, and summing the first and second terms and thenegative of the gradient of the cost function to generate the currentestimated image {circumflex over (m)}₁. However, since the second termis already known, a second iteration of the minimizing need only updatethe first term by multiplying the center mask W_(c) by the currentestimated image {circumflex over (m)}₁, and then sum the original secondterm with the updated first term arrive at the updated estimated image{circumflex over (m)}₂. Further, all subsequent iterations of theminimizing update the estimated image in a similar manner to the seconditeration. For example, all subsequent iterations re-use the second termand update the first term.

In one embodiment, the gradient of ƒ({circumflex over (m)})is calculatedas ∇ƒ({circumflex over (m)}), and the current estimated image{circumflex over (m)}_(i+1) is generated from a sum of the prior ithestimate and the negative of the gradient as shown by equation 3

{circumflex over (m)} _(i+1) =W _(p) {circumflex over (m)} _(i) +W _(c){circumflex over (m)} ₀ −W _(c)∇ƒ({circumflex over (m)}_(i))   (3)

The initial estimated image {circumflex over (m)}₀ may be given byequation 4 as follows:

{circumflex over (m)} ₀ℑ⁻¹ d   (4)

which shows that the initial estimated image {circumflex over (m)}₀ isthe inverse Fourier transform of gridded k-space data d. However, in analternate embodiment, the initial estimated image {circumflex over (m)}₀is the inverse Fourier transform of the k-space data. When imagingmodalities other than MRI are used, the inverse Fourier transform may bereplaced with a transform that is suitable for the correspondingmodality (e.g., back-projection for CT).

According to an exemplary embodiment of the invention, the cost functionof equation 1 is modified to a regional sparsity cost function as shownby the following equation 5.

ƒ({circumflex over (m)})=∥ℑ{circumflex over (m)}−d∥ ₂ +λ∥ΨW _(c){circumflex over (m)}∥ ₁   (5)

In the new cost function of equation 5, only the sparsity of the ROI ofthe estimated image {circumflex over (m)} is considered and calculated.For example, the ROI mask W_(c) is applied to (multiplied by) theestimated image in the second term of equation 1 to arrive at equation5.

Next, in the embodiment, the estimated image {circumflex over (m)} ofthe first term in equation 5 is replaced with {circumflex over(m)}=W_(c){circumflex over (m)}+W_(p){circumflex over (m)}₀ fromequation 2, and the gridded k-space data d is replaced with ℑ{circumflex over (m)}₀ since d=ℑ {circumflex over (m)}₀, which resultsin the following equation 6:

ƒ({circumflex over (m)})=∥ℑ(W _(c) {circumflex over (m)}+W _(p){circumflex over (m)} ₀)−ℑ{circumflex over (m)} ₀∥₂ +λ∥ΨW _(c){circumflex over (m)}∥ ₁.   (6)

Equation 6 can then be re-written as equation 7 when {circumflex over(m)}₀ is replaced with W_(c) {circumflex over (m)}₀+W_(p){circumflexover (m)}₀ as follows:

ƒ({circumflex over (m)})=∥ℑ(W _(c) {circumflex over (m)}+W _(p){circumflex over (m)} ₀)−ℑ(W _(c) {circumflex over (m)} ₀ +W _(p){circumflex over (m)} ₀)∥₂ +λ∥ΨW _(c) {circumflex over (m)}∥ ₁.   (7)

Equation 7 can then be simplified to equation 8 to generate the finalregional sparsity cost function as follows:

ƒ({circumflex over (m)})=∥ℑW _(c)({circumflex over (m)}−{circumflex over(m)} ₀)∥₂ +λ∥ΨW _(c) {circumflex over (m)}∥ ₁.   (8)

As shown by equation 8, CS is only applied on the ROI defined by the ROImask W_(c). Using this regional sparsity approach, CS reconstructionquality may be well-maintained and the time consumed by reconstructionmay be reduced. This approach can be used in real-time imagingapplications where the reconstruction time is desired to be less thanacquisition time for incoming real-time images to avoid delay betweenmeasurements.

Further, the regional sparsity approach may be extended to a CSreconstruction where only part of the images are considered sparse andare of interest in the final result. In equation (8), the ROI mask W_(c)can be generally defined as the region of interest of the image. Byapplying the mask, regional sparsity based CS reconstruction is onlyperformed on the ROI. This not only saves reconstruction time, but alsooffers the possibility to apply different sparsity to different regionson one image.

FIGS. 3 a-3 d illustrates a comparison between results obtained using aCS approach without masks and results obtained using a CS approachaccording to at least one embodiment of the invention (hereinafterreferred to as the “regional sparsity approach”). FIG. 3 a illustrates aresult of applying a CS calculation in both the ROI and “background”region according to the conventional approach.

FIG. 3 b illustrates a result of applying the regional sparsity approachby applying the CS calculation in only the ROI.

FIG. 3 c illustrates a difference image between images of FIG. 3 a andFIG. 3 b. As shown by FIG. 3 c the results achieved from using the CSapproach without masks differ minimally from the results achieved byusing the regional sparsity approach within the region of interest. FIG.3 d illustrates the difference image of FIG. 3 c magnified 10 times, toemphasis that the results from both approaches are quite similar withinROI.

FIG. 4 illustrates a method of reconstructing an image according to anexemplary embodiment of the invention. The method of FIG. 4 isconsidered a regional sparsity approach as well. The method of FIG. 4 isderived from the above equation 5. Referring to FIG. 4, the methodincludes acquiring raw image data (e.g., gridded or otherwise) during animage scan (e.g., MRI, CT, PET, etc.) of an area (S401). An image isthen estimated from the raw image data (S402). A difference is thencomputed between a transform of the estimated image and the raw imagedata (S403). If the modality is MRI, the transform can be Fourier, ifthe modality is CT, the transform can make use of back-projection, etc.The difference may be referred to as the fidelity term.

Next a background part of the estimated image is masked out (S404) and asparsity transform is applied to the masked out portion to yield asparse result (S405). The sparse result may be referred to as thesparsity term. The generation of the sparsity term can precede thegeneration of the fidelity term, or occur at the same or substantiallythe same time. Next, an image is reconstructed from a sum of thefidelity term and sparsity term (S406). For example, an image isreconstructed as the sum of the previous estimate and the negativegradient of the cost function. Blocks S403-S406 are then re-executed atleast once to update the estimated image based on prior estimates tooptimize the quality of the image that is ultimately reconstructed.

CS is an iterative process, which is very time consuming. The fidelityterm may contain a fast Fourier transform, which is typically quick. Thesparsity term contains a sparsity transform, which may be highlycomplex. The computation complexity depends on the amount of dataprocessed (e.g., the size of the image). If a smaller image can be used,less data needs to be processed, and thus any calculations performedthereon will be faster. At least one embodiment of the invention allowsthe sparsity transform to be performed only on the region of interest,which has a smaller size than the image. Further, by separating theimage into different regions, specific sparsities can be applied to theregions, which best represent the corresponding sparsity, and nocompromise is necessary. Accordingly, the sparsity level may beincreased and image reconstruction quality may be improved.

In a certain applications, the regional sparsity approach discussedabove may perform image reconstruction at least twice as fast as anyconventional approach, without significantly reducing image quality.

In a variation of at least one of the above-described embodiments, theestimated image is separated into more than one ROI and the backgroundregion. This variation is illustrated by FIG. 5, which shows the samesteps of S101 and S102 from FIGS. 1, and S103 and S104 have beenreplaced with S503 and S504. For example, the method includes separatingthe estimated image into several ROIs and a single background region(S503) and applying the CS with masks to iteratively update only theseveral ROIs and maintain the background region to reconstruct an image(S504). A separate and distinct ROI mask may be used to separate outeach of the several ROIs from the background region.

In an exemplary embodiment, when the estimated image has been separatedinto several ROIs, application of CS to update several ROIs may beperformed in a manner that uses different sparsity constraints. Forexample, the CS can apply one sparsity constraint to one of the severalROIs and another sparsity constraint to another one of the ROI. Examplesof these sparsity constraints include image domain sparsity, spatialtotal variance sparsity, wavelets sparsity, etc. For example, in oneembodiment, the CS uses image domain sparsity to update a first ROI andspatial total variance sparsity to update the second ROI.

FIG. 6 shows an example of a computer system, which may implementmethods and systems of the present disclosure. The system and methods ofthe present disclosure, or part of the system and methods, may beimplemented in the form of a software application running on a computersystem, for example, a mainframe, personal computer (PC), handheldcomputer, server, etc. For example, the method of FIGS. 1 and 4, and 5may be implemented as software application(s). These softwareapplications may be stored on a computer readable media (such as harddisk drive memory 1008) locally accessible by the computer system andaccessible via a hard wired or wireless connection to a network, forexample, a local area network, or the Internet.

The computer system referred to generally as system 1000 may include,for example, a central processing unit (CPU) 1001, a GPU (not shown), arandom access memory (RAM) 1004, a printer interface 1010, a displayunit 1011, a local area network (LAN) data transmission controller 1005,a LAN interface 1006, a network controller 1003, an internal bus 1002,and one or more input devices 1009, for example, a keyboard, mouse etc.As shown, the system 1000 may be connected to a data storage device, forexample, a hard disk, 1008 via a link 1007. CPU 1001 may be the computerprocessor that performs some or all of the steps of the methodsdescribed above with reference to FIGS. 1, 4, and 5.

Although the illustrative embodiments have been described herein withreference to the accompanying drawings, it is to be understood that thepresent invention is not limited to those precise embodiments, and thatvarious other changes and modifications may be affected therein by oneof ordinary skill in the related art without departing from the scope orspirit of the invention. All such changes and modifications are intendedto be included within the scope of the invention.

What is claimed is:
 1. A method for reconstructing an image comprises:estimating an image from acquired raw image data; separating theestimated image into a region of interest (ROI) and a background region;and applying compressed sensing to iteratively update only the ROI andmaintaining the background region to reconstruct an image.
 2. The methodof claim 1, wherein the background entirely surrounds the ROI.
 3. Themethod of claim 1, wherein the separating further separates theestimated image into a second ROI distinct from the first ROI, and theapplying comprises applying the compressed sensing to iteratively updatethe first ROI using a first sparsity constraint and iteratively updatethe second ROI using a second other sparsity constraint.
 4. The methodof claim 3, wherein one of the two sparsity constraints is image domainsparsity and the other is spatial total variance sparsity.
 5. The methodof claim 1, wherein prior to the estimating, the method comprises:acquiring un-gridded image data using a radial trajectory acquisitionand oversampling; and gridding the un-gridded image data to generate theraw image data.
 6. The method of claim 1, wherein estimating the imagecomprises performing an inverse Fourier transform on the raw image data.7. The method of claim 1, wherein the applying comprises performing thecompressed sensing on a sum of a fidelity term and a sparsity term,wherein the fidelity term includes a difference between a Fouriertransform of the estimated image and the raw image data, and wherein thesparsity term includes a mask multiplied by the estimated image thatmasks out the background region and retains the ROI.
 8. The method ofclaim 7, wherein the sparsity term applies a sparsity transform to aresult of the mask multiplied by the estimated image.
 9. The method ofclaim 8, wherein the sparsity transform generates a sparse result. 10.The method of claim 7, wherein the mask is a matrix of 1s and 0s, the 0scorresponding to a location of the background region and the 1scorresponding to a location of the ROI.
 11. The method of claim 7,wherein a weighting factor is multiplied by the sparsity term.
 12. Acomputer readable storage medium embodying instructions executable by aprocessor to perform method steps for reconstructing an image, themethod steps comprising instructions for: estimating an image fromacquired raw image data; separating the estimated image into a region ofinterest (ROI) and a background region; and applying compressed sensingto update only the ROI and maintain the background region to reconstructan image.
 13. The computer readable medium of claim 12, wherein thebackground region entirely surrounds the ROI.
 14. The computer readablestorage medium of claim 12, wherein the separating further separates theestimated image into a second ROI distinct from the first ROI, and theapplying comprises applying the compressed sensing to iteratively updatethe ROI using a first sparsity constraint and iteratively update thesecond ROI using a second other sparsity constraint.
 15. The computerreadable storage of claim 12, wherein the instructions further includeinstructions for: acquiring un-gridded image data using a radialtrajectory acquisition and oversampling; and gridding the un-griddedimage data to generate the raw image data.
 16. The computer readablestorage of claim 12, wherein the applying comprises performing thecompressed sensing on a sum of a fidelity term and a sparsity term,wherein the fidelity term includes a difference between a Fouriertransform of the estimated image and the raw image data, and wherein thesparsity term includes a mask multiplied by the estimated image thatmasks out the background region and retains the ROI.
 17. A method forreconstructing an image comprises: generating a cost function includinga fidelity term and a sparsity term; and minimizing the generated costfunction to generate a reconstructed image, wherein the fidelity termincludes a difference between a transform of an estimated image and rawimage data, and wherein the sparsity term includes a mask multiplied bythe estimated image that retains a region of interest (ROI) of theestimated image and masks out a background of the estimated image. 18.The method of claim 17, wherein the minimizing comprises performingcompressed sensing on the cost function.
 19. The method of claim 17,wherein prior to the generating of the cost function, the methodcomprises: acquiring un-gridded image data using a radial trajectoryacquisition and oversampling; and gridding the un-gridded image data togenerate the raw image data.
 20. The method of claim 17, wherein priorto the generating of the cost function, the method comprises performingan inverse Fourier transform on the raw image data to generate theestimated image.
 21. The method of claim 17, wherein the sparsity termapplies a sparsity transform to a result of the mask multiplied by theestimated image.
 22. The method of claim 17, wherein the backgroundregion entirely surrounds the ROI.